Both side of the Weight Painting went to the right side
To answer to my question if someone is stuck in the same situation:

The source of the problem:
I thought it was a bug but it's not. While editing my mesh I made a mistake: In Edit Mode, I used the Mesh/Symmetrize tool. I didn't think that it would not only mirror the position of the vertices, but their weights too! I believe using the Mirror Modifier instead would avoid this.

What's done is done, let's fix it!
In the Object Data Properties panel you can see all your Vertex Groups. They are assigned to bones by having the same name as them.



First, I made a copy of all the Vertex Groups that are inside of the right limbs. (Arm.R_copy)
Then, I erased the Vertex Groups in the left limbs. (Arm.L)
Finally, I renamed the copies, so they would assign to the left limb bones. (Arm.R_copy -&gt; Arm.L). So far, every Vertex Group has its weights, but also the opposite side. Let's get rid of that!
Select the LEFT half of your model. The easiest way to do it is to use lasso selection with X-Ray turned on (Alt+Z). Now, go through all the RIGHT Vertex Groups and click Remove, so the unecessary weights will be removed.
Select the RIGHT half, and do the same with the LEFT Vertex Groups.

-Peter